Multipliers, Paramultipliers, and weak-strong uniqueness for the Navier-Stokes equations

In this article, we describe spaces P such that : if u is a weak (in the sense of Leray [26]) solution of the Navier-Stokes system for some initial data u0, and if u belongs to P, then u is unique in the class of weak solutions. We say then that weak-strong uniqueness holds. It turns out that the proof of such results relies on the boundedness of a trilinear functional F : L2/αḢα × L2/βḢβ × P → R, where α, β belong to [0, 1]. In order to find optimal conditions for the boundedness of F , we are led to describing spaces of multipliers and of paramultipliers (that is, functions which map, by classical pointwise product or by paraproduct, a given Sobolev spaces in another given Sobolev space). The study of these spaces enables us to give conditions for weak-strong uniqueness which generalise all previously known results, from the famous Serrin criterion [41], to the recent conditions formulated by Lemarié [25].